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Question
Consider the situation of the previous problem. Define displacement resistance Rd = V/idof the space between the plates, where V is the potential difference between the plates and id is the displacement current. Show that Rd varies with time as `R_d = R(e^(t"/"tau) - 1)` .
Solution
Electric field strength for a parallel plate capacitor = `E = Q/(∈_0A)`
Electric flux , `phi = E . A = Q/(∈_0A).A = Q/∈_0`
Displacment current , `i_d = ∈_0 (dphi_E)/dt = ∈_0 d/dt(Q/∈_0)`
`i_d = ((dQ)/dt)`
Also , `Q = CV`
`i_d = d/dt(E_0C_e^(-t"/"RC))`
`i_d = E_0C(-1/(RC))e^-t"/"RC`
Displacement resistance , `R_d = E_0/i_d - R`
⇒ `R_d = E_0/(E_0/Re^(-t"/"RC)) - R`
⇒ `R_d = Re^(t"/"RC) - R`
⇒ `R_d = R(e^(t"/"RC) - 1)`
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